I am supposed to be using the fact that $e = \sum_{n=0} ^\infty \frac{1}{n!}$ to find the value of $\sum_{n=0} ^\infty \frac{1}{2n!}$.
Is there some method for substitution when dealing with infinite series, like how we have with arithmetic or functions? Every proof that I've seen for the above identity concerning $e$ is very specific for $e$, and I don't see how to provide a similar proof to find a value for $2n!$.
If you're allowed to use $e^x = \sum_{n = 0}^\infty \frac{x^n}{n!}$, then look at the expression for $e^1 + e^{-1}$ (assuming Lukas Geyer is correct in his comment above).